A crate starts from rest at the top of a large, frictionless spherical
surface, and slides into the water below. At what angle, θ, does
the crate leave the surface?
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To find the angle at which the crate leaves the surface, we can use the principles of conservation of energy and circular motion. Here's the step-by-step explanation of how to solve this question:
1. First, let's consider the initial and final states of the crate. Initially, the crate is at rest at the top of the spherical surface, and finally, it leaves the surface and falls into the water below.
2. Due to the absence of friction, the only forces acting on the crate are its weight and the normal force. As the crate starts sliding, it experiences a force component along the surface.
3. At the top of the spherical surface, the weight of the crate is acting vertically downward. The normal force cancels out the vertical component of the weight, leaving only the horizontal component of the weight as the net force acting on the crate. This horizontal force provides the necessary centripetal force to keep the crate moving in a circular path.
4. We can use the equation for centripetal force, F = m * (v^2)/r, where F is the net force, m is the mass, v is the velocity, and r is the radius of the circular path.
5. At the top of the spherical surface, the crate's velocity is zero. As it moves down the slope, it gains speed, and when it leaves the surface, its velocity is perpendicular to the surface. The radius of the circular path is the radius of the sphere.
6. At the top, the net force needed to keep the crate moving in a circle is provided solely by the horizontal component of the weight:
F = m * g * sin(θ) (1)
Where g is the acceleration due to gravity and θ is the angle at which the crate leaves the surface.
7. At the bottom, the net force is only the gravitational force acting vertically downward:
F = m * g (2)
8. To find the angle θ, we equate the expressions for the net force in equations (1) and (2):
m * g * sin(θ) = m * g
9. The mass cancels out, simplifying the equation:
sin(θ) = 1
10. Since sin(θ) equals 1 when θ = 90 degrees, the crate leaves the surface at an angle of 90 degrees or perpendicular to the surface.
Therefore, the crate leaves the surface at a 90-degree angle, or perpendicular to the surface.